The Relationship Between Permutations and Combinations
Simple Explanation
Permutations and combinations are directly related: P(n,r) = C(n,r) × r! — the number of ordered arrangements equals the number of unordered selections, multiplied by the r! ways each selected group could itself be ordered.
Why Do We Need It?
Understanding this relationship makes clear WHY the combinations formula divides by r! — and lets you convert between the two counts whenever needed.
Formula
The Relationship Between Permutations and Combinations
P(n,r) = C(n,r) × r!
The number of ordered arrangements (permutations) of r objects equals the number of unordered selections (combinations) of the same r objects, multiplied by the r! ways to order each selected group.
- P(n,r)
- — the number of ordered arrangements of r objects from n
- C(n,r)
- — the number of unordered selections of r objects from n
When to use it: Whenever converting between a permutation count and the corresponding combination count (or vice versa).
Worked Example
Convert a combination count into a permutation count
Given C(7,3)=35, find P(7,3) using the relationship between permutations and combinations.
Why Does This Work?
Every one of the C(n,r) unordered groups of r objects can itself be internally arranged in r! different orders — multiplying the number of groups by the number of orderings within each group counts every ordered arrangement exactly once, giving P(n,r) directly.
Real-Life Example
Converting between "who's on the team" and "who plays which position"
A coach first decides WHICH 3 players make the starting lineup (a combination), then decides WHO plays which of the 3 distinct positions (adding order).
The relationship P(n,r)=C(n,r)×r! directly connects these two related but different counts.
Practice
Given C(9,2)=36, find P(9,2).
MediumCommon mistake
Multiplying by n! or (n−r)! instead of r! when converting a combination count into the corresponding permutation count — only the r! (orderings WITHIN each selected group) belongs here.
Quick Review
- P(n,r) = C(n,r) × r!.
- r! accounts for the internal orderings within each selected group.
- This relationship explains exactly why the combinations formula divides by r!.